Calculus/Limits/Exercises

Basic Limit Exercises

1. $\lim _{x\to 2}{\Big [}4x^{2}-3x-1{\Big ]}$

2. $\lim _{x\to 5}{\Big [}x^{2}{\Big ]}$

3. $\lim _{x\to {\frac {\pi }{4}}}{\Big [}\cos ^{2}(x){\Big ]}$

4. $\lim _{x\to 1}{\Big [}5e^{x-1}-5{\Big ]}$

One-Sided Limits

Evaluate the following limits or state that the limit does not exist.

5. $\lim _{x\to 0^{-}}{\frac {x^{3}+x^{2}}{x^{3}+2x^{2}}}$

6. $\lim _{x\to 7^{-}}{\Big [}|x^{2}+x|-x{\Big ]}$

7. $\lim _{x\to -1^{+}}{\sqrt {1-x^{2}}}$

8. $\lim _{x\to -1^{-}}{\sqrt {1-x^{2}}}$

9. $\lim _{x\to 0^{-}}{\frac {|x|}{x}}$

10. $\lim _{x\to 1^{-}}\arcsin(x)$

Two-Sided Limits

Evaluate the following limits or state that the limit does not exist.

11. $\lim _{x\to -1}{\frac {1}{x-1}}$

12. $\lim _{x\to 4}{\frac {1}{x-4}}$

13. $\lim _{x\to 2}{\frac {1}{x-2}}$

14. $\lim _{x\to -3}{\frac {x^{2}-9}{x+3}}$

15. $\lim _{x\to 3}{\frac {x^{2}-9}{x-3}}$

16. $\lim _{x\to -1}{\frac {x^{2}+2x+1}{x+1}}$

17. $\lim _{x\to -1}{\frac {x^{3}+1}{x+1}}$

18. $\lim _{x\to 4}{\frac {x^{2}+5x-36}{x^{2}-16}}$

19. $\lim _{x\to 25}{\frac {x-25}{{\sqrt {x}}-5}}$

20. $\lim _{x\to 0}{\frac {|x|}{x}}$

21. $\lim _{x\to 2}{\frac {1}{(x-2)^{2}}}$

22. $\lim _{x\to 3}{\frac {\sqrt {x^{2}+16}}{x-3}}$

23. $\lim _{x\to -2}{\frac {3x^{2}-8x-3}{2x^{2}-18}}$

24. $\lim _{x\to 2}{\frac {x^{2}+2x+1}{x^{2}-2x+1}}$

25. $\lim _{x\to 3}{\frac {x+3}{x^{2}-9}}$

26. $\lim _{x\to -1}{\frac {x+1}{x^{2}+x}}$

27. $\lim _{x\to 1}{\frac {1}{x^{2}+1}}$

28. $\lim _{x\to 1}\left[x^{2}+5x-{\frac {1}{2-x}}\right]$

29. $\lim _{x\to 0}{\frac {x^{2}}{x^{2}+2x-3}}$

30. $\lim _{x\to 1}{\frac {x^{2}-1}{x^{2}+2x-3}}$

31. $\lim _{x\to 1}{\frac {5x}{x^{2}+2x-3}}$

32. $\lim _{x\to -1}\ln({\sqrt {x^{2}-1}})$

33. $\lim _{x\to 1}\arcsin(x)$

Limits to Infinity

Evaluate the following limits or state that the limit does not exist.

34. $\lim _{x\to \infty }{\frac {-x+\pi }{x^{2}+3x+2}}$

35. $\lim _{x\to -\infty }{\frac {x^{2}+2x+1}{3x^{2}+1}}$

36. $\lim _{x\to -\infty }{\frac {3x^{2}+x}{2x^{2}-15}}$

37. $\lim _{x\to -\infty }{\Big [}3x^{2}-2x+1{\Big ]}$

38. $\lim _{x\to \infty }{\frac {2x^{2}-32}{x^{3}-64}}$

39. $\lim _{x\to \infty }6$

40. $\lim _{x\to \infty }{\frac {3x^{2}+4x}{x^{4}+2}}$

41. $\lim _{x\to -\infty }{\frac {2x+3x^{2}+1}{2x^{2}+3}}$

42. $\lim _{x\to -\infty }{\frac {x^{3}-3x^{2}+1}{3x^{2}+x+5}}$

43. $\lim _{x\to \infty }{\frac {x^{2}+2}{x^{3}-2}}$

44. $\lim _{x\to \infty }{\frac {\sin \left({\frac {1}{x}}\right)}{x}}$

45. $\lim _{x\to -\infty }x^{2}\cos \left({\frac {1}{x}}\right)$

46. $\lim _{x\to \infty }{\frac {\sin(\arctan(x))}{\arctan(-x)}}$

Limits of Piecewise Functions

Evaluate the following limits or state that the limit does not exist.

48. Consider the function

$f(x)={\begin{cases}(x-2)^{2}&{\text{if }}x<2\\x-3&{\text{if }}x\geq 2\end{cases}}$

a. $\lim _{x\to 2^{-}}f(x)$

b. $\lim _{x\to 2^{+}}f(x)$

c. $\lim _{x\to 2}f(x)$

49. Consider the function

$g(x)={\begin{cases}-2x+1&{\text{if }}x\leq 0\\x+1&{\text{if }}0<x<4\\x^{2}+2&{\text{if }}x\geq 4\end{cases}}$

a. $\lim _{x\to 4^{+}}g(x)$

b. $\lim _{x\to 4^{-}}g(x)$

c. $\lim _{x\to 0^{+}}g(x)$

d. $\lim _{x\to 0^{-}}g(x)$

e. $\lim _{x\to 0}g(x)$

f. $\lim _{x\to 1}g(x)$

50. Consider the function

$h(x)={\begin{cases}2x-3&{\text{if }}x<2\\8&{\text{if }}x=2\\-x+3&{\text{if }}x>2\end{cases}}$

a. $\lim _{x\to 0}h(x)$

b. $\lim _{x\to 2^{-}}h(x)$

c. $\lim _{x\to 2^{+}}h(x)$

d. $\lim _{x\to 2}h(x)$

Intermediate Value Theorem

51. Use the intermediate value theorem to show that there exists a value $f(x)=3$ for $f(x)={\sqrt {x}}+{\frac {1}{\sqrt {x}}}$ from $1\leq x\leq 9$. If you cannot use the intermediate value theorem to show this, explain why.

External Links

• Limits 1 exercise at Khan Academy
• Limits 2 exercise at Khan Academy